Home > Trigonometry > Proving Identities

How to verify 2csc^2 (x) = ((1/(1-sinx))+((1/(1+sinx))?

Answer 1

Grace Ashcraft

#2sec^2x#

#"using the "color(blue)"trigonometric identities"#

#•color(white)(x)sin^2x+cos^2x=1#

#•color(white)(x)secx=1/cosx#

#"consider the right side"#

#1/(1-sinx)+1/(1+sinx)#

#"expressing as a single fraction "#

#(1+sinx+1-sinx)/((1-sinx)(1+sinx))#

#=2/(1-sin^2x)#

#=2/cos^2x=2sec^2x!=2csc^2x#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Brooklyn Anderson

To verify the identity ( 2 \csc^2(x) = \frac{1}{1 - \sin(x)} + \frac{1}{1 + \sin(x)} ):

Start with the left-hand side:

[ 2 \csc^2(x) = 2 \left(\frac{1}{\sin^2(x)}\right) ]

Now, find a common denominator for the fractions on the right-hand side:

[ \frac{1}{1 - \sin(x)} + \frac{1}{1 + \sin(x)} ]

[ = \frac{1 + 1 - \sin(x) + 1 + \sin(x)}{(1 - \sin(x))(1 + \sin(x))} ]

[ = \frac{2}{1 - \sin^2(x)} ]

[ = 2 \csc^2(x) ]

Therefore, the left-hand side is equal to the right-hand side, and the identity is verified.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.